Let $S$ be a regular semigroup and $E(S)$ be the set of its idempotents. We call the sets $S(e,f)f$ and $eS(e,f)$ one-sided sandwich sets and characterize them abstractly where $e,f \in E(S)$. For $a, a^{\prime } \in S$ such that $a=aa^{\prime }a$, $a^{\prime }=a^{\prime }aa^{\prime }$, we call $S(a)=S(a^{\prime }a, aa^{\prime })$ the sandwich set of $a$. We characterize regular semigroups $S$ in which all $S(e,f)$ (or all $S(a))$ are right zero semigroups (respectively are trivial) in several ways including weak versions of compatibility of the natural order. For every $a \in S$, we also define $E(a)$ as the set of all idempotets $e$ such that, for any congruence $\rho $ on $S$, $a \rho a^2$ implies that $a \rho e$. We study the restrictions on $S$ in order that $S(a)$ or $E(a)\cap D_{a^2}$ be trivial. For $\mathcal F \in \lbrace \mathcal S, \mathcal E\rbrace $, we define $\mathcal F$ on $S$ by $a \mathrel {\mathcal F}b$ if $F(a) \cap F (b)\ne \emptyset $. We establish for which $S$ are $\mathcal S$ or $\mathcal E$ congruences.
For any two positive integers n and k\geqslant 2, let G(n, k) be a digraph whose set of vertices is {0, 1, ..., n − 1} and such that there is a directed edge from a vertex a to a vertex b if ak ≡ b (mod n). Let n = \prod\nolimits_{i = 1}^r {p_i^{{e_i}}} be the prime factorization of n. Let P be the set of all primes dividing n and let P_{1},P_{2} \subseteq P be such that P_{1\cup P_{2}}= P and P_{1\cup P_{2}}=\emptyset . A fundamental constituent of G(n, k), denoted by G_{{P_2}}^*(n,k), is a subdigraph of G(n, k) induced on the set of vertices which are multiples of \prod\nolimits_{{p_i} \in {P_2}} {{p_i}} and are relatively prime to all primes q\in P_{1}. L. Somer and M. Křižek proved that the trees attached to all cycle vertices in the same fundamental constituent of G(n, k) are isomorphic. In this paper, we characterize all digraphs G(n, k) such that the trees attached to all cycle vertices in different fundamental constituents of G(n, k) are isomorphic. We also provide a necessary and sufficient condition on G(n, k) such that the trees attached to all cycle vertices in G(n, k) are isomorphic., Amplify Sawkmie, Madan Mohan Singh., and Obsahuje seznam literatury
We investigate some (universal algebraic) properties of residuated lattices—algebras which play the role of structures of truth values of various systems of fuzzy logic.
For a subalgebra ${\mathcal B}$ of a partial monounary algebra ${\mathcal A}$ we define the quotient partial monounary algebra ${\mathcal A}/{\mathcal B}$. Let ${\mathcal B}$, ${\mathcal C}$ be partial monounary algebras. In this paper we give a construction of all partial monounary algebras ${\mathcal A}$ such that ${\mathcal B}$ is a subalgebra of ${\mathcal A}$ and ${\mathcal C}\cong {\mathcal A}/{\mathcal B}$.
We develop a general axiomatic theory of algebraic pairs, which simultaneously generalizes several algebraic structures, in order to bypass negation as much as feasible. We investigate several classical theorems and notions in this setting including fractions, integral extensions, and Hilbert's Nullstellensatz. Finally, we study a notion of growth in this context.
We assign to each pair of positive integers $n$ and $k\ge 2$ a digraph $G(n,k)$ whose set of vertices is $H=\{0,1,\dots ,n-1\}$ and for which there is a directed edge from $a\in H$ to $b\in H$ if $a^k\equiv b\pmod n$. We investigate the structure of $G(n,k)$. In particular, upper bounds are given for the longest cycle in $G(n,k)$. We find subdigraphs of $G(n,k)$, called fundamental constituents of $G(n,k)$, for which all trees attached to cycle vertices are isomorphic.